I'm trying to reproduce the following solution to the ODE, $$\frac{d^{2}u}{d\rho^{2}} = \frac{l*(l+1)}{\rho^{2}}u$$
Solution: $$u\left(\rho\right) = C\rho^{l+1}+D\rho^{-l}$$
What I've tried in Mathematica using DSolve
is
DSolve[u''[ρ] == (l*(l + 1))/ρ^2*u[ρ], u[ρ], ρ]
However, when I try using Simplify
or FullSimplify
on my Mathematica output, I can't get it into the simple version as above.
The Mathematica Output Solution that I get from the use of DSolve
is
{{u[ρ] -> ρ^(1/2 I Sqrt[l] Sqrt[1 + l] (-(I/(Sqrt[l] Sqrt[1 + l])) - Sqrt[-4 - 1/(l (1 + l))]))C[1]
+ ρ^(1/2 I Sqrt[l] Sqrt[1 + l] (-(I/(Sqrt[l] Sqrt[1 + l])) + Sqrt[-4 - 1/(l (1 + l))]))C[2]}}
Thanks very much for your valuable time.